A particle P starts from the point z 0 = 1 + 2 i , where i = - 1 . It moves horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z 1 . From z 1 the particle moves 2 units in the direction of th

Question

A particle $P$ starts from the point $z_{0} = 1 + 2 i$ , where $i = \sqrt{- 1}$ . It moves horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_{1}$ . From $z_{1}$ the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i} + \hat{j}$ and then it moves through an angle $\frac{π}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point $z_{2}$ . The point $z_{2}$ is given by [2008]

Smart Achievers · Accepted Answer

(4)

$The initial position of point is Z_{0} = 1 + 2 i$

$∴$ $Z_{1} = (1 + 5) + (2 + 3) i = 6 + 5 i$

$z_{2}$

$∴$ $It becomes Z_{1}' = Z_{1} + (1 + i) = 7 + 6 i$

$Now O Z_{1}' is rotated through an angle \frac{π}{2} in anticlockwise direction,$

$therefore Z_{2} = i Z_{1}' = - 6 + 7 i .$

Solution

1 $6 + 7 i$

2 $- 7 + 6 i$

3 $7 + 6 i$

4 $- 6 + 7 i$

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Solution

1 6+7i

2 -7+6i

3 7+6i

4 -6+7i

Ans. (4) The initial position of point is Z0=1+2i ∴ Z1=(1+5)+(2+3)i=6+5i Now Z1 is moved through a distance of 2 units in the direction i^+j^ (i.e. by 1+i) ∴ It becomes Z1'=Z1+(1+i)=7+6i Now OZ1' is rotated through an angle π2 in anticlockwise direction, therefore Z2=iZ1'=-6+7i.

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1 $6 + 7 i$

2 $- 7 + 6 i$

3 $7 + 6 i$

4 $- 6 + 7 i$